Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove that if $g=0$, then the weak solution of \begin{align} -\Delta u = f,& \text{in }\Omega\\ \frac{\partial u}{\partial \nu} = g,&\text{on }\Gamma \end{align} is actually in $H^2$. If $g\neq 0$, is $u$ still in $H^2$? If so, is there a simple proof? (Of course, suppose that f and g are consistent.)

I found paper by Umezu [2] where the case with $C^\infty$ boundary is treated in a more general setting.

References

[1] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1985.

[2] K. Umezu, $L^p$-Approach to Mixed Boundary Value Problems for Second-Order Elliptic Operators, Tokyo J. Math. Vol 17, No. 1, 1994.